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If $X$ is a collection of sets, define $\bigcap X$ = $\bigcap_{U \in X} U$. According to this definition,

$$\bigcap \emptyset \ne \emptyset$$

Too see this, note that if $x \notin \bigcap \emptyset$, there exists $U \in \emptyset$ such that $x \notin U$. However, by definition, that's impossible.

But what set would $\bigcap \emptyset$ be? Everything?

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